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Polynomial Control Systems (2014)

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4.1.1 The Smith Form

The Smith form of a polynomial matrix, left matrix-fraction, right matrix-fraction, or system matrix object, can be determined by using the function SmithForm.

Finding the Smith form.

Make sure the application is loaded.

In[1]:=

Here is a square polynomial matrix in the variable s.

In[2]:=
Out[3]=

This is the Smith form of this polynomial matrix.

In[4]:=
Out[5]=

Here is a system matrix in the variable s.

In[6]:=
Out[7]=

This is the Smith form of this system matrix.

In[8]:=
Out[9]=

When analyzing matrix-fraction objects, the designer is often interested in the Smith form of the combination of the numerator and denominator matrices, namely, the composite matrix for a left matrix fraction or the composite matrix for a right matrix fraction, as explained later in Section 4.3 on coprimeness.

Here is a left matrix-fraction model of a system with two inputs and three outputs.

In[10]:=
Out[11]=

This is the Smith form of the matrix pair [D(s) N(s)] of this left matrix fraction.

In[12]:=
Out[13]=

Since the resulting Smith form is not equal to [I3 0], the two matrices D(s) and N(s) describing the left matrix-fraction object are not relatively left prime. This means that a lower-order model of the left matrix fraction exists that will generate the same transfer-function model, as explained later in Section 4.3.